I have a list of C n=35, k=5 combinations. Each 5-tuple in every combination has 4 differences: tuple/combination "7 12 23 11 33" - I sort it first and then get 4 differences: "7 11 12 23 33" ==> 7-12, 11-12, 12-23, 23-33. Let's speak about absolute value of those differences.
Now I want to know, which four differences maximize number of combinations. Say, all combinations with four differences being 1,2,3,4 give 12% of all C combinations.
How can it be done?
I checked (just counted by computer) that different sets of differences produce different number of respective combinations, so there should be optimal solution!
Since you are talking about a sequence (of $4$ differences), I assume order matters. I.e. $(1,2,3,4)$ is different from $(1,3,2,4)$. Then given a sequence $(a,b,c,d)$, the only allowed subsets are $(x, x+a, x+a+b, x+a+b+c, x+a+b+c+d)$ for some $x$. The only constraints are $x \ge 1$ and $x+a+b+c+d \le 35$. E.g. for the sequence $(1,4,2,3), x$ can be anywhere from $1$ to $25$. So the number of such subsets $= 35 - (a+b+c+d)$. The number of subsets is maximized for $(1,1,1,1)$.
OTOH, if (despite your choice of the word "sequence") order does not matter, i.e. $(1,2,3,4)$ can be permuted into $(1,3,2,4)$ etc., then: for each specific ordering the above logic counts the number of subsets, but different orderings gives different subsets. So the number of subsets $= (35 - (a+b+c+d)) \times $ no. of ways to order $(a,b,c,d)$ (which is $4! = 24$ if they are all different). The number of subsets is maximized for $(1,2,3,4)$.